The origin of diffraction limit in wave physics, and the way to overcome it, can be revisited using the time-reversal mirror concept. According to time-reversal symmetry, a broadband wave can be focused both in time and space regardless of the complexity of a scattering medium. In a complex environment a time-reversal mirror acts as an antenna that uses complex environments to appear wider than it is, resulting in a refocusing quality that does not depend on the time-reversal antenna aperture.  The broadband nature of time-reversed waves distinguishes them from continuous phase-conjugated waves and allows revisiting the origin of diffraction limits, suggesting new ways to obtained subwavelength focusing for broadband waves.

One approach consists in introducing the initial source inside a micro structured medium made of subwavelength resonators with a mean distance smaller than the used wavelengths. It will be shown that, for a broadband source located inside such structure, a time-reversal mirror located in the far field radiated a time-reversed wave that interacts with the medium (random or periodic) to regenerate not only the propagating but also the evanescent waves required to refocus below the diffraction limit. This focusing process is very different from the one developed with superlenses made of negative index material only valid for narrowband signals.  We will emphasize the role of the frequency diversity in time-reversal focusing and a modal description of the spatiotemporal focusing will be presented. It shows the super-resolution properties obtained with acoustic and electromagnetic waves suggesting for the future also new kind of metamaterials for optical waves.

Another approach is related to the concept of a perfect time-reversal experiment that needs, not only to time-reverse the wavefield but also to time-reverse the source. It is the concept of an acoustic or electromagnetic “sink” or drain that is related to the perfect absorber theory. Is it possible to build a blackbody of infinitively small size? 

For a field of definition $k$ of an abelian variety $\Av$ and prime ideal $\ip$ of $k$ which is of a good reduction for $\Av$, the structure of $\Av(\F_{\ip})$ as abelian group is:

    \Av(\F_{\ip})\simeq \Z/d_1(\ip)\Z\oplus\cdots\oplus\Z/d_g(\ip)\Z\oplus\Z/e_1(\ip)\Z\oplus\cdots\oplus\Z/e_g(\ip)\Z,

    where $d_i(\ip)|d_{i+1}(\ip)$, $d_g(\ip)|e_1(\ip)$, and $e_i(\ip)|e_{i+1}(\ip)$ for $1\leq i<g$.

    We are interested in finding an asymptotic formula for the number of prime ideals $\ip$ with $N\ip<x$, $\Av$ has a good reduction at $\ip$, $d_1(\ip)=1$. We succeed in this under the assumption of the Generalized Riemann Hypothesis (GRH). Unconditionally, we achieve a short range asymptotic for abelian varieties of CM type, and the full cyclicity theorem for elliptic curves over a number field containing CM field.